Moderation 調節 = 交互作用
|
|
- Tyler Cooper
- 5 years ago
- Views:
Transcription
1 Moderation 調節 = 交互作用 Kit-Tai Hau 侯傑泰 JianFang Chang 常建芳 The Chinese University of Hong Kong Based on Marsh, H. W., Hau, K. T., Wen, Z., Nagengast, B., & Morin, A. J. S. (in press). Moderation. In Little, T. D. (Ed.), Oxford Handbook of Quantitative Methods. New York: Oxford University Press. Chinese Psychological Society -13-th Conference, Shanghai 2010
2 Introduction Examples Ed Psych: effects of an instructional technique interact with students characteristics Dev Psych: effects of a variable interact with age Soc Psych: effects of individual characteristic depends on Group Organizational Psych: employee characteristics workplace characteristics Moderator: variable affects direction and/or strength of relation between indep var and dep var, typically defined as X 1 X 2 (Moderation = interaction) mediation Latent Interaction -- Marsh, Wen, Nagengast, Hau 2
3 Mediation (MED) vs Moderation (MOD) X = predictor variable Y = outcome variable INT = interaction A: no mediation or moderation B: effect of X on Y mediated in part by MED; total mediation if direct effect of X on Y is zero (β1=0) or partial medication (if β1 0); indirect effect of X on Y = α 1 β 2 C, D: represent interaction effect; MOD moderates the relation between X and Y
4 Introduction Moderator: variable that affects the direction and/or strength of relationship between independent (predictor) variable and dependent (outcome) variable when/for whom does X has a stronger/ weaker relation/effect on Y? Moderation Hau & Chang 4
5 Introduction Traditional (nonlatent) Approaches Interaction between two manifest variables (X 1, X 2 ) on outcome (Y) X 1, X 2 small number of categories: ANOVA X 1, X 2 cont., regression to estimate main and int n Y = β + β X + β X + β X X + = β Helpful to graph if interaction is significant Empirical interactions typically small, non-sig, substantial measurement error reduces power of sig test Latent interaction controls for measurement error, increase power, provide more defensible interpretation of interaction e Latent Interaction -- Marsh, Wen, Nagengast, Hau 5
6 Graphs of Interactions inter n sig Graph to understand nature Moderation Hau & Chang 6
7 Categorical Variables: Analysis of Variance (ANOVA) Null: neither predictor depends on value of other tempt to transform continuous variables into ANOVA approach (M split into high/low group) Problems: (i) reduce reliability (loss power), (ii) reduce var explained by original variables, (iii) no summary estimate of strength of interaction, (iv) difficult to detect non linear effects Transform only when categorization is natural (e.g., minimal passing score) Summary: almost never transform cont var -> categories for ANOVA Moderation Hau & Chang 7
8 Separate Group Multiple Regression One indep var is categorical (few levels, e.g., gender), another indep var continuous tempted to use separate group regression Inter n = differences between unstd regression coefficients, possible sig test for 2 groups Weakness: not facilitate interpretation of effects (inter n sig? uncertain); reduce power due to small N in each group; one IV must be truly categorical can use the more general approach Moderation Hau & Chang 8
9 Moderated Multiple Regression Approach Y = b 0 + b 1 X 1 + b 2 X 2 + e effect of X 2 moderated by X 1 Y = (b 0 + b 1 X 1 ) + (b 2 + b 3 X 1 ) X 2 Y = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 1 X 2 + e standardized Z β + β Z + β Z + Z Z + e Y = Z 0 Z1 X 1 Z 2 X 2 β Z 3 X 1 X 2 β z1 = change in Y (in SD unit) if X 1 change 1 SD at X 2 =0 Regression plots represent predicted values based on the model (not raw data) More than 2 groups -> dummy/effect coding Moderation Hau & Chang 9 Z
10 Moderated Multiple Regression - 2 Use meaningful zero point; typically at 0, ±1 (or 2) SD Moderation Hau & Chang 10
11 Moderated Multiple Regression - 3 Effects of X 1 and X 2 on Y are not unconditional main effect; depends on values of other variable X 1, X 2 seldom take 0 (or arbitrary) regression wt on centered and std variables often more useful Std coefficients not straightforward from commercial stat packages; to obtain: Std (z-score) all variables Z Y, Z X1, Z X2 form interaction term = Z X1 Z X2 (but not re-std) Predict Z Y, with Z X1, Z X2, Z X1 Z X2, use t-values, and report unstd coef as appropriate std solutions Moderation Hau & Chang 11
12 Moderated Multiple Regression - 4 regression models testing the interaction term must contain the main effects of predictor variables (X 1,X 2 ) for categorical var involving more then 2 dummy var, all related product terms entered simultaneously, test sig of change in R 2 with/without interaction terms covariates (gender, age) generally added as 1 st set of variables (but check rationale) Regions of significant (range of values of moderator in which simple slopes are significant can be plotted (see Preacher et al J Ed Beh Stat 31, ) Moderation Hau & Chang 12
13 Moderated Multiple Regression - 5 Interaction Point: for disordinal (crossing) interaction, the intersection point for X 1 as the moderator, X 2 = b 1 /b 3 Power in detecting interactions: difficult in finding substantially meaningful, statistically significant interactions because Overall model errors in non-expt studies generally larger than controlled expt Measurement are exaggerated when multiplied to form prouducts Moderation Hau & Chang 13
14 Moderated Multiple Regression - 6 Magnitude of inter n constrained in field studies when researchers cannot assign participants to optional levels of predictor variables Detect inter n compromised by nonlinearitie of effects To have sufficient power: Large N Similar N across subgroups Avoid restriction of range High reliability scales Moderation Hau & Chang 14
15 Two Broad categories Latent Variable Approaches at least one variable involved is categorical with few categories (e.g., male/female) multiple group SEM both variables are continuous and latent various approaches and best practice still evolving Latent Interaction -- Marsh, Wen, Nagengast, Hau 15
16 Latent Variable Approach Multiple Group Analysis latent variable (ξ 1 ) observed categorical variable (X 2 ) latent variable (η) X 2 small number of naturally existing categories, as grouping var test: invariance of ξ 1 η effects over multiple groups; decline in goodness of fit with invariance constraint easily implemented in most SEM software problems: limitation in interpretation of the interaction, reduce power (small N), ignore measurement error categorizing var Not recommended, unless it is a true catergorial var with small number of categories with at least moderate sample sizes Latent Interaction -- Marsh, Wen, Nagengast, Hau 16
17 Latent Variable Approaches Full Latent (variable) Approach Kenny & Judd (1984) proposed an ingenious heuristic model by constraining of loadings/variances of the product term η γ ξ + γ ξ + γ ξ ξ + = ζ. Latent Interaction -- Marsh, Wen, Nagengast, Hau 17
18 Latent Variable Approach Main Issues different ways to form the product indicator; How many product indicators? How to form best set? many constraints on parameters make the method tedious /difficulty, are they absolutely necessary? even if both ξ 1 ξ 2 have mean of zero, product term ξ 1 ξ 2 mean is not zero; mean structure complicates the application, is it really necessary? typical software do not provide appropriate SE for std effects, more serious with interaction model, how to obtain appropriate std solution? Latent Interaction -- Marsh, Wen, Nagengast, Hau 18
19 Parameter Constrained & UnconstrainedApproaches Constrained Approach Kenny & Judd (1984) proposed an ingenious heuristic model by constraining of loadings/variances of the product term η γ ξ + γ ξ + γ ξ ξ + = ζ. x =ξ + 2 = λ 2ξ1 + δ δ1 x x3 = ξ 2 + δ3 4 4ξ2 δ4 x =λ + Latent Interaction -- Marsh, Wen, Nagengast, Hau 19
20 Strategies for Creating Product Indicators 2 guidelines use all the information (all multiple indicators should be used in forming product indictors) do NOT reuse information: each multiple indicator used once in forming product indicators to avoid artificially created correlated residuals (variance/covariance matrix of errors becomes diagonal) Other possible strategies Use the better indicators (when cannot use all indicators) Use parcels (average of indicators) when there are too many indicators in a certain indep var Latent Interaction -- Marsh, Wen, Nagengast, Hau 20
21 Parameter Constrained & UnconstrainedApproaches constrained approach (cont) Judd suggested using x 1 x 3, x 1 x 4,x 2 x 3,x 2 x 4 as indicators of the interaction ξ 1 ξ 2 and imposed many constraints on loadings and variances, e.g. x x = λ λ4ξ1ξ 2 λ2ξ1δ 4 λ4ξ 2δ2 δ2δ4 Marsh et al.(2002) simulation showed: unconstrained approach is recommended for its ease in implementation and acceptable bias /precision Summary: mean-center all x, y indicators, create product indicator, fit model without mean structure (because software routinely centers them again) Latent Interaction -- Marsh, Wen, Nagengast, Hau 21
22 ,, An Appropriate Standardized Solution and Its Scale-free Properties (cont) appropriate std solution of interaction model not directly provided by usual commercial software Wen, Marsh, Hau (2010) derived appropriate std solution for latent interaction, which are scale free, SE and t-values are also scale free Let usual std coefficients be γ, γ 1 2, γ 3, appropriate std coefficients γ 1, γ, γ are obtained: γ = γ 1 γ 2 = γ 2 γ 3 = γ 3 φ 33 φ = φ = ξ ) where φ 11 = var( ξ 1 ) 22 var( ξ 2 ) are from the original solutions φ φ var( ξ Latent Interaction -- Marsh, Wen, Nagengast, Hau 22
23 An Appropriate Standardized Solution and Its Scale-free Properties (cont) Scale-free properties of std solution Wen, Marsh, Hau (2010) proved that the appropriate std estimates have the scale-free properties invariant when calcualted from either the centered or std data Calculation of SE of appropriate std coef through Bootstrap samples (similar to original estimates) t-values of original estimates can be used to test the significance of the appropriate std estimates, if close to cutoff point use bootstrap method Latent Interaction -- Marsh, Wen, Nagengast, Hau 23
24 =0.197;. Unconstrained Approaches: Examples η = γ ξ + γ ξ + γ ξ ξ ( ξ ξ )] + ζ [ 1 2 E 1 2 Each latent variable has 3 indicators Assume η is math achievement, ξ 1 is prior math ability,ξ 2 is math motivation, ξ 1 ξ 2 is the interaction of prior math ability and math motivation x x x x x y 1 to y 3, x 1 x 6 centered, product indicators x 1 4,, are created, but not re-standardized γ 1=0.425, γ 2 =0.331, γ 3 =0.197; φ =0.501, φ =0.529, φ =0.308; and the completely standardized estimates: γ 1 =0.423, γ =0.338 and γ 2 3 = By using Formula 27, γ 1 =0.423, γ 2 =0.338, and. γ = 0. Latent Interaction -- Marsh, Wen, Nagengast, Hau 24
25 Unconstrained Approaches: Examples (cont) Path diagram and estimates (constrained model Latent Interaction -- Marsh, Wen, Nagengast, Hau 25
26 Robustness to Normality in Unconstrained Approach Considerations when normality is violated: ML typically used is based on assumption of normality, however, this is a common problem to all CFA, SEM research (not specific to interaction/quadratic analyses) even when ξ 1,ξ 2 are normal, the product are non-normal, constrained, partially constrained, unconstrained all suffer when ML estimation is used Fortunately, ML tends to be robust to violation of normality in parameter estimates, though ML likelihood ratio test is too large, standard errors are too small under nonnormality Latent Interaction -- Marsh, Wen, Nagengast, Hau 26
27 Distribution-analytic Approaches Whereas they have many desirable features, they are computationally demanding, and not available in widely accessible SEM softwares Latent Moderated Structural Equation (LMS, Klein & Moosbrugger, 2000) implemented in Mplus Quasi-Maximum Likelihood(QML, Klein & Muthén, 2002) available from author, not available in software yet QML (Klein & Muthen, 2002) was developed for more efficient estimation than LMS Both estimate parameters in η = α + Γ ξ + ξ Ωξ + ζ LMS and QML differ in the distributional assumptions about the latent dependent variable η and its indicators Computationally LMS is more efficient and can be used to fit models with a larger number of nonlinear effects and Latent Interaction -- Marsh, Wen, Nagengast, Hau 27 interactions
28 Summary One of predictor variables is a manifest grouping variable with small number of categories multiple group SEM, but not recommended when all predictors are continuous or based on multiple indicators Product-indicator dominated latent interaction research, still evolving, unconstrained approach ease of implementation and robustness More recently, LMS/QML hold considerable promise over product-indicator approach Many issues not appropriately dealt with and applied research is limited Latent Interaction -- Marsh, Wen, Nagengast, Hau 28
29 Thank You
What is in the Book: Outline
Estimating and Testing Latent Interactions: Advancements in Theories and Practical Applications Herbert W Marsh Oford University Zhonglin Wen South China Normal University Hong Kong Eaminations Authority
More informationABSTRACT. Chair, Dr. Gregory R. Hancock, Department of. interactions as a function of the size of the interaction effect, sample size, the loadings of
ABSTRACT Title of Document: A COMPARISON OF METHODS FOR TESTING FOR INTERACTION EFFECTS IN STRUCTURAL EQUATION MODELING Brandi A. Weiss, Doctor of Philosophy, 00 Directed By: Chair, Dr. Gregory R. Hancock,
More informationLatent variable interactions
Latent variable interactions Bengt Muthén & Tihomir Asparouhov Mplus www.statmodel.com November 2, 2015 1 1 Latent variable interactions Structural equation modeling with latent variable interactions has
More informationEstimation of Curvilinear Effects in SEM. Rex B. Kline, September 2009
Estimation of Curvilinear Effects in SEM Supplement to Principles and Practice of Structural Equation Modeling (3rd ed.) Rex B. Kline, September 009 Curvlinear Effects of Observed Variables Consider the
More informationReview of Statistics 101
Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods
More informationSC705: Advanced Statistics Instructor: Natasha Sarkisian Class notes: Introduction to Structural Equation Modeling (SEM)
SC705: Advanced Statistics Instructor: Natasha Sarkisian Class notes: Introduction to Structural Equation Modeling (SEM) SEM is a family of statistical techniques which builds upon multiple regression,
More informationReview of Multiple Regression
Ronald H. Heck 1 Let s begin with a little review of multiple regression this week. Linear models [e.g., correlation, t-tests, analysis of variance (ANOVA), multiple regression, path analysis, multivariate
More informationModule 3. Latent Variable Statistical Models. y 1 y2
Module 3 Latent Variable Statistical Models As explained in Module 2, measurement error in a predictor variable will result in misleading slope coefficients, and measurement error in the response variable
More informationGlobal Model Fit Test for Nonlinear SEM
Global Model Fit Test for Nonlinear SEM Rebecca Büchner, Andreas Klein, & Julien Irmer Goethe-University Frankfurt am Main Meeting of the SEM Working Group, 2018 Nonlinear SEM Measurement Models: Structural
More informationOutline
2559 Outline cvonck@111zeelandnet.nl 1. Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) 1.1 Similarities and differences between MRA with dummy variables
More informationSpecifying Latent Curve and Other Growth Models Using Mplus. (Revised )
Ronald H. Heck 1 University of Hawai i at Mānoa Handout #20 Specifying Latent Curve and Other Growth Models Using Mplus (Revised 12-1-2014) The SEM approach offers a contrasting framework for use in analyzing
More informationResearch Design - - Topic 19 Multiple regression: Applications 2009 R.C. Gardner, Ph.D.
Research Design - - Topic 19 Multiple regression: Applications 2009 R.C. Gardner, Ph.D. Curve Fitting Mediation analysis Moderation Analysis 1 Curve Fitting The investigation of non-linear functions using
More informationRuth E. Mathiowetz. Chapel Hill 2010
Evaluating Latent Variable Interactions with Structural Equation Mixture Models Ruth E. Mathiowetz A thesis submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment
More informationDesigning Multilevel Models Using SPSS 11.5 Mixed Model. John Painter, Ph.D.
Designing Multilevel Models Using SPSS 11.5 Mixed Model John Painter, Ph.D. Jordan Institute for Families School of Social Work University of North Carolina at Chapel Hill 1 Creating Multilevel Models
More informationMultilevel Modeling: A Second Course
Multilevel Modeling: A Second Course Kristopher Preacher, Ph.D. Upcoming Seminar: February 2-3, 2017, Ft. Myers, Florida What this workshop will accomplish I will review the basics of multilevel modeling
More informationCourse Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model
Course Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 1: August 22, 2012
More informationEstimation of nonlinear latent structural equation models using the extended unconstrained approach
Review of Psychology, 009, Vol. 6, No., 3-3 UDC 59.9 Estimation of nonlinear latent structural equation models using the extended unconstrained approach AUGUSTIN KELAVA HOLGER BRANDT In the past two decades,
More informationCourse Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model
Course Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model EPSY 905: Multivariate Analysis Lecture 1 20 January 2016 EPSY 905: Lecture 1 -
More informationFormula for the t-test
Formula for the t-test: How the t-test Relates to the Distribution of the Data for the Groups Formula for the t-test: Formula for the Standard Error of the Difference Between the Means Formula for the
More informationStructural Equation Modeling and Confirmatory Factor Analysis. Types of Variables
/4/04 Structural Equation Modeling and Confirmatory Factor Analysis Advanced Statistics for Researchers Session 3 Dr. Chris Rakes Website: http://csrakes.yolasite.com Email: Rakes@umbc.edu Twitter: @RakesChris
More informationCentering Predictor and Mediator Variables in Multilevel and Time-Series Models
Centering Predictor and Mediator Variables in Multilevel and Time-Series Models Tihomir Asparouhov and Bengt Muthén Part 2 May 7, 2018 Tihomir Asparouhov and Bengt Muthén Part 2 Muthén & Muthén 1/ 42 Overview
More informationExploring Cultural Differences with Structural Equation Modelling
Exploring Cultural Differences with Structural Equation Modelling Wynne W. Chin University of Calgary and City University of Hong Kong 1996 IS Cross Cultural Workshop slide 1 The objectives for this presentation
More information36-309/749 Experimental Design for Behavioral and Social Sciences. Dec 1, 2015 Lecture 11: Mixed Models (HLMs)
36-309/749 Experimental Design for Behavioral and Social Sciences Dec 1, 2015 Lecture 11: Mixed Models (HLMs) Independent Errors Assumption An error is the deviation of an individual observed outcome (DV)
More informationMplus Code Corresponding to the Web Portal Customization Example
Online supplement to Hayes, A. F., & Preacher, K. J. (2014). Statistical mediation analysis with a multicategorical independent variable. British Journal of Mathematical and Statistical Psychology, 67,
More informationChapter 5. Introduction to Path Analysis. Overview. Correlation and causation. Specification of path models. Types of path models
Chapter 5 Introduction to Path Analysis Put simply, the basic dilemma in all sciences is that of how much to oversimplify reality. Overview H. M. Blalock Correlation and causation Specification of path
More informationA Re-Introduction to General Linear Models (GLM)
A Re-Introduction to General Linear Models (GLM) Today s Class: You do know the GLM Estimation (where the numbers in the output come from): From least squares to restricted maximum likelihood (REML) Reviewing
More informationConfirmatory Factor Analysis: Model comparison, respecification, and more. Psychology 588: Covariance structure and factor models
Confirmatory Factor Analysis: Model comparison, respecification, and more Psychology 588: Covariance structure and factor models Model comparison 2 Essentially all goodness of fit indices are descriptive,
More informationMultilevel Structural Equation Modeling
Multilevel Structural Equation Modeling Joop Hox Utrecht University j.hox@uu.nl http://www.joophox.net 14_15_mlevsem Multilevel Regression Three level data structure Groups at different levels may have
More informationSPECIAL TOPICS IN REGRESSION ANALYSIS
1 SPECIAL TOPICS IN REGRESSION ANALYSIS Representing Nominal Scales in Regression Analysis There are several ways in which a set of G qualitative distinctions on some variable of interest can be represented
More informationPlausible Values for Latent Variables Using Mplus
Plausible Values for Latent Variables Using Mplus Tihomir Asparouhov and Bengt Muthén August 21, 2010 1 1 Introduction Plausible values are imputed values for latent variables. All latent variables can
More informationNesting and Equivalence Testing
Nesting and Equivalence Testing Tihomir Asparouhov and Bengt Muthén August 13, 2018 Abstract In this note, we discuss the nesting and equivalence testing (NET) methodology developed in Bentler and Satorra
More informationGeneral structural model Part 1: Covariance structure and identification. Psychology 588: Covariance structure and factor models
General structural model Part 1: Covariance structure and identification Psychology 588: Covariance structure and factor models Latent variables 2 Interchangeably used: constructs --- substantively defined
More informationMANOVA is an extension of the univariate ANOVA as it involves more than one Dependent Variable (DV). The following are assumptions for using MANOVA:
MULTIVARIATE ANALYSIS OF VARIANCE MANOVA is an extension of the univariate ANOVA as it involves more than one Dependent Variable (DV). The following are assumptions for using MANOVA: 1. Cell sizes : o
More informationFactor Analysis. Qian-Li Xue
Factor Analysis Qian-Li Xue Biostatistics Program Harvard Catalyst The Harvard Clinical & Translational Science Center Short course, October 7, 06 Well-used latent variable models Latent variable scale
More informationLatent Variable Centering of Predictors and Mediators in Multilevel and Time-Series Models
Latent Variable Centering of Predictors and Mediators in Multilevel and Time-Series Models Tihomir Asparouhov and Bengt Muthén August 5, 2018 Abstract We discuss different methods for centering a predictor
More informationSupplementary materials for: Exploratory Structural Equation Modeling
Supplementary materials for: Exploratory Structural Equation Modeling To appear in Hancock, G. R., & Mueller, R. O. (Eds.). (2013). Structural equation modeling: A second course (2nd ed.). Charlotte, NC:
More informationOne-Way ANOVA. Some examples of when ANOVA would be appropriate include:
One-Way ANOVA 1. Purpose Analysis of variance (ANOVA) is used when one wishes to determine whether two or more groups (e.g., classes A, B, and C) differ on some outcome of interest (e.g., an achievement
More informationIntroducing Generalized Linear Models: Logistic Regression
Ron Heck, Summer 2012 Seminars 1 Multilevel Regression Models and Their Applications Seminar Introducing Generalized Linear Models: Logistic Regression The generalized linear model (GLM) represents and
More informationSTAT 730 Chapter 9: Factor analysis
STAT 730 Chapter 9: Factor analysis Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Data Analysis 1 / 15 Basic idea Factor analysis attempts to explain the
More informationCorrelation and simple linear regression S5
Basic medical statistics for clinical and eperimental research Correlation and simple linear regression S5 Katarzyna Jóźwiak k.jozwiak@nki.nl November 15, 2017 1/41 Introduction Eample: Brain size and
More informationAlternatives to Difference Scores: Polynomial Regression and Response Surface Methodology. Jeffrey R. Edwards University of North Carolina
Alternatives to Difference Scores: Polynomial Regression and Response Surface Methodology Jeffrey R. Edwards University of North Carolina 1 Outline I. Types of Difference Scores II. Questions Difference
More informationRon Heck, Fall Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October 20, 2011)
Ron Heck, Fall 2011 1 EDEP 768E: Seminar in Multilevel Modeling rev. January 3, 2012 (see footnote) Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October
More informationInference using structural equations with latent variables
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationA Study of Statistical Power and Type I Errors in Testing a Factor Analytic. Model for Group Differences in Regression Intercepts
A Study of Statistical Power and Type I Errors in Testing a Factor Analytic Model for Group Differences in Regression Intercepts by Margarita Olivera Aguilar A Thesis Presented in Partial Fulfillment of
More informationChapter 4: Factor Analysis
Chapter 4: Factor Analysis In many studies, we may not be able to measure directly the variables of interest. We can merely collect data on other variables which may be related to the variables of interest.
More informationRegression: Main Ideas Setting: Quantitative outcome with a quantitative explanatory variable. Example, cont.
TCELL 9/4/205 36-309/749 Experimental Design for Behavioral and Social Sciences Simple Regression Example Male black wheatear birds carry stones to the nest as a form of sexual display. Soler et al. wanted
More informationSupplemental material for Autoregressive Latent Trajectory 1
Supplemental material for Autoregressive Latent Trajectory 1 Supplemental Materials for The Longitudinal Interplay of Adolescents Self-Esteem and Body Image: A Conditional Autoregressive Latent Trajectory
More informationComputationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models
Computationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models Tihomir Asparouhov 1, Bengt Muthen 2 Muthen & Muthen 1 UCLA 2 Abstract Multilevel analysis often leads to modeling
More informationInteraction effects for continuous predictors in regression modeling
Interaction effects for continuous predictors in regression modeling Testing for interactions The linear regression model is undoubtedly the most commonly-used statistical model, and has the advantage
More information36-309/749 Experimental Design for Behavioral and Social Sciences. Sep. 22, 2015 Lecture 4: Linear Regression
36-309/749 Experimental Design for Behavioral and Social Sciences Sep. 22, 2015 Lecture 4: Linear Regression TCELL Simple Regression Example Male black wheatear birds carry stones to the nest as a form
More informationClass Notes: Week 8. Probit versus Logit Link Functions and Count Data
Ronald Heck Class Notes: Week 8 1 Class Notes: Week 8 Probit versus Logit Link Functions and Count Data This week we ll take up a couple of issues. The first is working with a probit link function. While
More informationCategorical and Zero Inflated Growth Models
Categorical and Zero Inflated Growth Models Alan C. Acock* Summer, 2009 *Alan C. Acock, Department of Human Development and Family Sciences, Oregon State University, Corvallis OR 97331 (alan.acock@oregonstate.edu).
More informationConsequences of measurement error. Psychology 588: Covariance structure and factor models
Consequences of measurement error Psychology 588: Covariance structure and factor models Scaling indeterminacy of latent variables Scale of a latent variable is arbitrary and determined by a convention
More informationMeasurement Error. Martin Bland. Accuracy and precision. Error. Measurement in Health and Disease. Professor of Health Statistics University of York
Measurement in Health and Disease Measurement Error Martin Bland Professor of Health Statistics University of York http://martinbland.co.uk/ Accuracy and precision In this lecture: measurements which are
More informationPath Analysis. PRE 906: Structural Equation Modeling Lecture #5 February 18, PRE 906, SEM: Lecture 5 - Path Analysis
Path Analysis PRE 906: Structural Equation Modeling Lecture #5 February 18, 2015 PRE 906, SEM: Lecture 5 - Path Analysis Key Questions for Today s Lecture What distinguishes path models from multivariate
More informationANOVA, ANCOVA and MANOVA as sem
ANOVA, ANCOVA and MANOVA as sem Robin Beaumont 2017 Hoyle Chapter 24 Handbook of Structural Equation Modeling (2015 paperback), Examples converted to R and Onyx SEM diagrams. This workbook duplicates some
More informationActivity #12: More regression topics: LOWESS; polynomial, nonlinear, robust, quantile; ANOVA as regression
Activity #12: More regression topics: LOWESS; polynomial, nonlinear, robust, quantile; ANOVA as regression Scenario: 31 counts (over a 30-second period) were recorded from a Geiger counter at a nuclear
More informationRonald Heck Week 14 1 EDEP 768E: Seminar in Categorical Data Modeling (F2012) Nov. 17, 2012
Ronald Heck Week 14 1 From Single Level to Multilevel Categorical Models This week we develop a two-level model to examine the event probability for an ordinal response variable with three categories (persist
More informationApplication of Plausible Values of Latent Variables to Analyzing BSI-18 Factors. Jichuan Wang, Ph.D
Application of Plausible Values of Latent Variables to Analyzing BSI-18 Factors Jichuan Wang, Ph.D Children s National Health System The George Washington University School of Medicine Washington, DC 1
More informationReview of CLDP 944: Multilevel Models for Longitudinal Data
Review of CLDP 944: Multilevel Models for Longitudinal Data Topics: Review of general MLM concepts and terminology Model comparisons and significance testing Fixed and random effects of time Significance
More informationSEM Day 3 Lab Exercises SPIDA 2007 Dave Flora
SEM Day 3 Lab Exercises SPIDA 2007 Dave Flora 1 Today we will see how to estimate SEM conditional latent trajectory models and interpret output using both SAS and LISREL. Exercise 1 Using SAS PROC CALIS,
More informationMeasurement Invariance Testing with Many Groups: A Comparison of Five Approaches (Online Supplements)
University of South Florida Scholar Commons Educational and Psychological Studies Faculty Publications Educational and Psychological Studies 2017 Measurement Invariance Testing with Many Groups: A Comparison
More informationOnline Appendix for Sterba, S.K. (2013). Understanding linkages among mixture models. Multivariate Behavioral Research, 48,
Online Appendix for, S.K. (2013). Understanding linkages among mixture models. Multivariate Behavioral Research, 48, 775-815. Table of Contents. I. Full presentation of parallel-process groups-based trajectory
More informationCHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA
Examples: Multilevel Modeling With Complex Survey Data CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA Complex survey data refers to data obtained by stratification, cluster sampling and/or
More informationØ Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.
Statistical Tools in Evaluation HPS 41 Fall 213 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific
More informationComparing IRT with Other Models
Comparing IRT with Other Models Lecture #14 ICPSR Item Response Theory Workshop Lecture #14: 1of 45 Lecture Overview The final set of slides will describe a parallel between IRT and another commonly used
More informationAn Introduction to Path Analysis
An Introduction to Path Analysis PRE 905: Multivariate Analysis Lecture 10: April 15, 2014 PRE 905: Lecture 10 Path Analysis Today s Lecture Path analysis starting with multivariate regression then arriving
More informationChapter 8: Regression Models with Qualitative Predictors
Chapter 8: Regression Models with Qualitative Predictors Some predictors may be binary (e.g., male/female) or otherwise categorical (e.g., small/medium/large). These typically enter the regression model
More informationStructural equation modeling
Structural equation modeling Rex B Kline Concordia University Montréal ISTQL Set B B1 Data, path models Data o N o Form o Screening B2 B3 Sample size o N needed: Complexity Estimation method Distributions
More informationTABLE OF CONTENTS INTRODUCTION TO MIXED-EFFECTS MODELS...3
Table of contents TABLE OF CONTENTS...1 1 INTRODUCTION TO MIXED-EFFECTS MODELS...3 Fixed-effects regression ignoring data clustering...5 Fixed-effects regression including data clustering...1 Fixed-effects
More informationScore-Based Tests of Measurement Invariance with Respect to Continuous and Ordinal Variables
Score-Based Tests of Measurement Invariance with Respect to Continuous and Ordinal Variables Achim Zeileis, Edgar C. Merkle, Ting Wang http://eeecon.uibk.ac.at/~zeileis/ Overview Motivation Framework Score-based
More informationModule 2. General Linear Model
D.G. Bonett (9/018) Module General Linear Model The relation between one response variable (y) and q 1 predictor variables (x 1, x,, x q ) for one randomly selected person can be represented by the following
More informationDaniel Boduszek University of Huddersfield
Daniel Boduszek University of Huddersfield d.boduszek@hud.ac.uk Introduction to moderator effects Hierarchical Regression analysis with continuous moderator Hierarchical Regression analysis with categorical
More informationExperimental Design and Data Analysis for Biologists
Experimental Design and Data Analysis for Biologists Gerry P. Quinn Monash University Michael J. Keough University of Melbourne CAMBRIDGE UNIVERSITY PRESS Contents Preface page xv I I Introduction 1 1.1
More informationSEM Day 1 Lab Exercises SPIDA 2007 Dave Flora
SEM Day 1 Lab Exercises SPIDA 2007 Dave Flora 1 Today we will see how to estimate CFA models and interpret output using both SAS and LISREL. In SAS, commands for specifying SEMs are given using linear
More informationAn Introduction to Multilevel Models. PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012
An Introduction to Multilevel Models PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012 Today s Class Concepts in Longitudinal Modeling Between-Person vs. +Within-Person
More informationDimensionality Assessment: Additional Methods
Dimensionality Assessment: Additional Methods In Chapter 3 we use a nonlinear factor analytic model for assessing dimensionality. In this appendix two additional approaches are presented. The first strategy
More informationSEM for Categorical Outcomes
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationMLMED. User Guide. Nicholas J. Rockwood The Ohio State University Beta Version May, 2017
MLMED User Guide Nicholas J. Rockwood The Ohio State University rockwood.19@osu.edu Beta Version May, 2017 MLmed is a computational macro for SPSS that simplifies the fitting of multilevel mediation and
More informationIntroduction to Linear regression analysis. Part 2. Model comparisons
Introduction to Linear regression analysis Part Model comparisons 1 ANOVA for regression Total variation in Y SS Total = Variation explained by regression with X SS Regression + Residual variation SS Residual
More informationNew developments in structural equation modeling
New developments in structural equation modeling Rex B Kline Concordia University Montréal Set B: Mediation A UNL Methodology Workshop A2 Topics o Mediation: Design requirements Conditional process modeling
More informationIntroduction to Matrix Algebra and the Multivariate Normal Distribution
Introduction to Matrix Algebra and the Multivariate Normal Distribution Introduction to Structural Equation Modeling Lecture #2 January 18, 2012 ERSH 8750: Lecture 2 Motivation for Learning the Multivariate
More informationA Threshold-Free Approach to the Study of the Structure of Binary Data
International Journal of Statistics and Probability; Vol. 2, No. 2; 2013 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education A Threshold-Free Approach to the Study of
More informationNon-linear panel data modeling
Non-linear panel data modeling Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini May 2010 Laura Magazzini (@univr.it) Non-linear panel data modeling May 2010 1
More informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS Page 1 MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level
More informationGrowth Mixture Model
Growth Mixture Model Latent Variable Modeling and Measurement Biostatistics Program Harvard Catalyst The Harvard Clinical & Translational Science Center Short course, October 28, 2016 Slides contributed
More informationConfidence Intervals for a Semiparametric Approach to Modeling Nonlinear Relations among Latent Variables
Structural Equation Modeling, 18:537 553, 2011 Copyright Taylor & Francis Group, LLC ISSN: 1070-5511 print/1532-8007 online DOI: 10.1080/10705511.2011.607072 Confidence Intervals for a Semiparametric Approach
More informationAdditional Notes: Investigating a Random Slope. When we have fixed level-1 predictors at level 2 we show them like this:
Ron Heck, Summer 01 Seminars 1 Multilevel Regression Models and Their Applications Seminar Additional Notes: Investigating a Random Slope We can begin with Model 3 and add a Random slope parameter. If
More informationModel Estimation Example
Ronald H. Heck 1 EDEP 606: Multivariate Methods (S2013) April 7, 2013 Model Estimation Example As we have moved through the course this semester, we have encountered the concept of model estimation. Discussions
More informationFall Homework Chapter 4
Fall 18 1 Homework Chapter 4 1) Starting values do not need to be theoretically driven (unless you do not have data) 2) The final results should not depend on starting values 3) Starting values can be
More informationGeneralized Linear. Mixed Models. Methods and Applications. Modern Concepts, Walter W. Stroup. Texts in Statistical Science.
Texts in Statistical Science Generalized Linear Mixed Models Modern Concepts, Methods and Applications Walter W. Stroup CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint
More informationImpact of serial correlation structures on random effect misspecification with the linear mixed model.
Impact of serial correlation structures on random effect misspecification with the linear mixed model. Brandon LeBeau University of Iowa file:///c:/users/bleb/onedrive%20 %20University%20of%20Iowa%201/JournalArticlesInProgress/Diss/Study2/Pres/pres.html#(2)
More informationMediation question: Does executive functioning mediate the relation between shyness and vocabulary? Plot data, descriptives, etc. Check for outliers
Plot data, descriptives, etc. Check for outliers A. Nayena Blankson, Ph.D. Spelman College University of Southern California GC3 Lecture Series September 6, 2013 Treat missing i data Listwise Pairwise
More informationADVANCED C. MEASUREMENT INVARIANCE SEM REX B KLINE CONCORDIA
ADVANCED SEM C. MEASUREMENT INVARIANCE REX B KLINE CONCORDIA C C2 multiple model 2 data sets simultaneous C3 multiple 2 populations 2 occasions 2 methods C4 multiple unstandardized constrain to equal fit
More informationInvestigating Models with Two or Three Categories
Ronald H. Heck and Lynn N. Tabata 1 Investigating Models with Two or Three Categories For the past few weeks we have been working with discriminant analysis. Let s now see what the same sort of model might
More informationBasic IRT Concepts, Models, and Assumptions
Basic IRT Concepts, Models, and Assumptions Lecture #2 ICPSR Item Response Theory Workshop Lecture #2: 1of 64 Lecture #2 Overview Background of IRT and how it differs from CFA Creating a scale An introduction
More informationTesting Main Effects and Interactions in Latent Curve Analysis
Psychological Methods 2004, Vol. 9, No. 2, 220 237 Copyright 2004 by the American Psychological Association 1082-989X/04/$12.00 DOI: 10.1037/1082-989X.9.2.220 Testing Main Effects and Interactions in Latent
More informationRegularized Common Factor Analysis
New Trends in Psychometrics 1 Regularized Common Factor Analysis Sunho Jung 1 and Yoshio Takane 1 (1) Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1, Canada
More informationReview of the General Linear Model
Review of the General Linear Model EPSY 905: Multivariate Analysis Online Lecture #2 Learning Objectives Types of distributions: Ø Conditional distributions The General Linear Model Ø Regression Ø Analysis
More informationCorrelation and Regression Bangkok, 14-18, Sept. 2015
Analysing and Understanding Learning Assessment for Evidence-based Policy Making Correlation and Regression Bangkok, 14-18, Sept. 2015 Australian Council for Educational Research Correlation The strength
More information